On the AVD-total chromatic number of circulant graphs

Abstract

AVD-k-total coloring of a simple graph G is a mapping π : V (G) ∪ E(G) → {1, . . ., k} such that: adjacent or incident elements x, y ∈ V (G) ∪ E(G), π(x) ≠ π(y); and for each pair of adjacent vertices x, y ∈ V (G), sets {π(x)} ∪ {π(xv) | xv ∈ E(G) and v ∈ V (G)} and {π(y)} ∪ {π(yv) | yv ∈ E(G) and v ∈ V (G)} are distinct. The AVD-total chromatic number, denoted by χ′′a(G) is the smallest k for which G admits an AVD-k-total-coloring. [Zhang et al. 2005] conjectured that any graph G has χ′′a(G) ≤ ∆+3. [Hulgan 2009] conjectured that any subcubic graph G has χ′′a(G) ≤ 5. In this article, we proved that all cubic circulant graph has χ′′a(C2n(d, n))) = 5, being positive evidence to Hulgan’s conjecture.